Group Definition (expanded) - Abstract Algebra

[@Socratica]

If you'd like to learn more, we have a free course on Group Theory! www.socratica.com/courses/group-theory

[@elgs1980]

A solid proof how important a good teacher is!

[@davidmeng2840]

Exactly. The video explains everything clearly!

[@studiohq]

SOLID PROOF!

[@pajeetsingh]

Easy on that Qian Chen

[@eldeee143]

I'd really agree with you on this. My professor literally just went on to blabber mathematical shit that I didn't really know of but this video and the teacher made all the difference. Hands up

[@atul.binda.mithlesh]

yes truly indeed, for the first time ever I found studying abstract algebra can also be fun.

[@NervousNoodles]

Me: It's time to go to bed. KZbin: _True but have you seen first "Group Definition (expanded) - Abstract Algebra"?_

[@vonstreckerz]

Same thing here !

[@mohammedriyas669]

Same here

[@violinpracice6440]

same lol and I enjoyed

[@ibrahimshafi2064]

Lol 2am here

[@SameerKhan-nd5qb]

Same here Xd

[@stancartmankenny]

In college, I was amazed how one of the first major theorems you learn is the fundamental theorem of calculus, and how the last one you learn is the fundamental theorem of arithmetic.

[@agentsmidt3209]

I was talking to a math major the other day. I am just a lay man, but I could remember Logic and sets were taught way further down (or not taught at all) in Uni math classes. The complete reverse...no wonder math is so confusing to many.

[@RadicalCaveman]

Well, that's because the "Fundamental Theorem of Arithmetic" is really the fundamental theorem of number theory. On the other hand, they COULD teach number theory sooner. It's not that hard, but it's not requisite for everything else the way calculus is.

[@MauricioMartinez0707]

Because calculus requires you to build on top of your current understanding of math, while the the fundamental theorem of arithmetic requires you to rebuild your current understanding of math. the second one requires a much more advanced mindset to actually understand

[@Tanuvloghamirpur002]

Jjhkjo

[@insouciantFox]

@@RadicalCaveman People complain now that "I never use math", can you imagine what those people would say if they taught *number theory* instead of (I dont know) algebra?

[@toomuchcandor3293]

I can't believe how this made hours of my confusion just vanish away in mere minutes. Truly thank you for helping me begin to see the beauty of this concept

[@hellfirelordofevil]

This is the most well made course on the KZbin by a significant margin

[@GPS509]

I am a spanish teacher. Sometimes I also help students in High school with Math. Thus, I am looking forward to improve my skill in this very important field.

[@massielzayas1967]

Math and spanish, mi respeto para usted

[@nyoongggmmgg9825]

@Rafael Arturo Mateo Núñez math is for anyone dude

[@lerimatdu93]

@Rafael Arturo Mateo Núñez they litteraly did during the bourbaki period in France, 12-14 years kids confronted to basics of abstract an linear algebra.. It worked wonders on ~10% of them and it was an utter disaster for the rest. Totally reversed now.

[@goalkeeper5755]

I passed an exam on Modern Algebra thanks to your playlist Abstract Algebra, which helps a lot to imagine these boring theorems and formulas. Thank you and please continue doing so magnificent lectures!

[@jason-mr]

There are not boring, at least not for me.

[@almspb]

I am here just to express my gratitude as already valuable and well structured comments have been made regarding how you have made easier to understand "Group Definition" for us, who're having difficulties to understand this. Thank you very much and my sincere, well wishes to your channel!

[@Socratica]

Thank you so much for your kind message! Much appreciated!! 💜🦉

[@m4riel]

It's amazing how a new perspective can totally change the way you view and deal with a subject. For months I'm trying to make sense of how different groups and their operations really work, and this video has made a huuuge difference. I'm very grateful for your job.

[@AadarshLokhande]

I don't know how much time taken by mathematicians to define a group, but I only knew that it took me only 8 min to understand. Purely Simplified ❤

[@nathanlaney1542]

Extremely well crafted video; you're a great speaker!

[@kantaprasadsinha8025]

Good lesson on higher Math.

[@mrugankthatte4378]

The presenter was so calm and composed. Watching this video felt like meditation to me.

[@frankx8739]

So nice to have something explained so clearly. I once went though a book on Group Theory where it failed to tell you what it was.

[@hmeyer827]

I'm so glad to have found this video. I'm teaching a survey/liberal arts class in mathematics, and students are required to do math history presentations. One student is presenting on Galois but needs help in understand what Group Theory is (at the most basic level). This is a FANTASTIC introduction - short and accessible and accurate. I will be sharing this video with him. Thank you!

[@Socratica]

Thank you so much for sharing with your student!! We're rooting for his success! Be well, Socratica Friend! 💜🦉

[@tododiaissobicho]

My algebra teacher made it seem completely random and arbitrary when talking about this subject. These videos made everything SO MUCH CLEARER

[@Socratica]

We're so glad you're watching Socratica! We have a whole series of videos about Abstract Algebra: bit.ly/AbstractAlgebraSocratica Subscribe here to see our latest videos: bit.ly/SocraticaSubscribe

[@s.k.potdarpotdar8377]

bye socratica. i am your fan and admirer.when i watch your videos i fell as if godess saraswati is dancing on your tongue

[@brianambelis5869]

thank you so very much for creating this video. it was so clear and easy to follow. I've been wanting to learn abstract algebra but didn't know where to start. I'm gonna go watch all your other videos on abstract algebra. do you have any recommendations for textbooks for self study on the subject? i think it would help me supplement your lectures to get an even deeper understanding. thanks

[@tarunsenpradhani1534]

Thank you very much ma'am

[@PeterParker-zk1yh]

A day will come when i will be a good teacher like her Her teaching method is amazing

[@dipakbanerjee6236]

@@s.k.potdarpotdar8377 4

[@Ayaz-Edits]

Thanks Alot ,its really Helpful 💗💗

[@supersaiyan-goku-san]

This blew my mind! Precise and to the point. Plus also be able to give the general overview! Excellent.

[@grrsa]

I'm taking a mathematics for machine learning course. I understood everything until they brought up groups. They provided a very academic source explaining groups which I barely understood. This video made it so easy to understand. THANK YOU!

[@zoso25]

This is a scintillating instance of great pedagogy! One just requires pretty basic conceptual background to grasp this video in its entirety. I'll support you... Please keep doing this. I'm grateful KZbin put this in my list.I was missing out on such great content.

[@amitygames9318]

I don't know how I ended up here after watching a Miley Cirus video, but it feels like landing on a fresh, clean, wholesome, normal planet .... Awesome ! Keep it up.

[@wahabfiles6260]

"wholesome".. lol... millenials!

[@bp56789]

You came in like a wrecking ball. We Can't Stop judging you.

[@deathvall3y]

love the Miley Cyrus part

[@dohpam1ne]

Fantastic video. The structure and delivery of the material is great. It's incredible how intuitive this topic can be if it's taught properly.

[@gN0mseryiops]

Her voice is so soothing, but not in a way that puts me to sleep. Its clear and well supported, what a beautiful sound. I'm currently an undergrad studying biology, I've only taken beginner calculus and statistics, but I am considering a minor in mathematics. These videos are definitely a big inspiration, as well as having a great teacher.

[@jeffwilken7241]

Your delivery is just...flawless.

[@fahrenheit2101]

God, why isnt this the first course KZbin recommends on group theory, it's leagues clearer than the others. You've earned a sub

[@matrixate]

Great production quality. Hiring an actress, writing a script, and producing the visuals....that costs time and money. I hope you guys can keep it up!

[@iitmotivationwithrahullson5930]

You are the best.. I mean your teaching style is totally different but most important is that we understood your lecture... India..

[@paulallen579]

These videos rock! I’m crying, that’s how elegant, informative and funny this content is! Bless! ❤️

[@north_state]

I opened a book on algebra and within half an hour closed it because I was unable to understand nothing. But saw these videos and the mystery unfolds....now I know how the universe works😊.THANK YOU.

[@deacondank6249]

Ok... I usually get weird recommended from KZbin, but this just takes it to a whole other level.

[@ihabsultan1]

This is called 'recommendation system', what it really means is that folks who you hang with all the time betrayed you and went to study math behind your back.

[@al470ex12]

Proccecing decades of math from an 11:14 video definatly made me appreciate the legends, it also activated the fire alarm above my head. Good video really helpfull to learn these before starting my semester.

[@WesleyTheGLEEK]

You are an extremely great teacher! This channel needs to be recognized af

[@InvaderMixo]

So at 11:00 the implication is that she is *just* on the verge of going full Heisenberg, and the only thing stopping her is her devotion to making AMAZING math videos like this one. Superb!

[@pramod120895]

Saw the video when randomly scrolling down. But now I am in the excitement of learning group theory deeply. Thanks a lot. From INDIA

[@m.sherif-deenyahaya5998]

In less than 1 hour i have learn more from than ever on group theory. Thanks a lot .

[@lexx8335]

Can't believe I've been struggling so much when this page actually existed. Sincerely from the bottom of my heart THANK YOU!

[@Socratica]

We're SO glad when we hear our videos are helping people. It really inspires us to keep making more videos! 💜🦉

[@sabinaixchelgarciamendoza]

Please don't stop making this wonderfull videos i mean WOAH! never seen abstract algebra so interesting THANK YOU!

[@martyglacerda]

You are awesome. I’m taking abstract algebra and discrete math (combinatorics and graph theory) this semester. It’s videos like these that convinced me to switch my major to mathmematics a year ago. I will be sure to make a contribution. -Marty (University of Victoria, Canada)

[@MacCionnaith]

This is really nice to watch. I was always drawn towards math. I studied engineering in university which is very practical, but then when it comes to work it's mostly basic arithmatic and algebra. I miss mathematics as a pure subject 😒.

[@PopeLando]

That was the best introduction to Groups I've ever seen. Others concentrate too much on its relationship with symmetry. This showed simply how general Groups are. And the other introduction I saw called the 1st rotation "r120". It's actually wrong and confusing to use a number, the degrees of rotation, to explain the process. But the one rotation to turn the triangle and leave it looking the same is r, and then doing it again is r². Because she started straight away with "doing nothing is 1," you could jump ahead and see that r³=1 before she said it. Great teaching!

[@philomath6190]

Algebra is the abstraction of arithmetic. Thus, abstract algebra is the abstraction of the abstraction of arithmetic. 😁

[@benhur2806]

What about the abstraction of the abstraction of the abstraction of the abstraction of the abstraction of the abstraction of the abstraction of the abstraction of abstract algebra? ;-)

[@imaginaryunit.]

I guess that would be category theory. en.wikipedia.org/wiki/Category_theory

[@vibodhj349]

@@benhur2806 Abstractception Algebra.

[@MrAntAve]

That'll keep me awake, thanks man

[@RadicalCaveman]

So abstract algebra is to arithmetic as acceleration is to distance.

[@ghazalehrahimi432]

that was by far THE BEST explanation video that I've ever seen! Can't believe these sort of contents are free on youtube! Thank you so much!

[@vaishnav_mallya]

This is awesome. Math taught the way it's meant to be.

[@MrOvipare]

For some reason, my spectroscopy professor could'nt explain group theory in such a clear fashion. What a great channel!

[@smartcat001]

Great video, very clearly explained the subject. Thanks, I learned something.

[@yash1152]

1:59 _"mod is short for modular"_ thanks a lot for this, "modulo" never clearered the ambiguity between modulus (the absolute) and mod (the now more apt, modular)

[@ANGEL1999PATO]

Been missing the abstract algebra videos, thanks :)

[@prasadnastik9239]

Mam.....just take a bow.... Clear explanation..... Love ur teaching..... Best math tutorial.... I had ever seen...... Thankyou very much mam....

[@RaviSoni-om2nh]

Great video series! Can you please make a video on Lie Group and Lie Algebra?

[@codewithimad]

I'm really speechless of how good of a teacher you're thank you for doing this

[@gabrielgmj]

I feel so smart after watching this, thank you for the self-esteem boost ❤️

[@learnmathematicswithmuhamm9243]

How you use animation in your lecture

[@alllkoy]

I'm so thankfull for this material! I'm freshmen in uni and currently we're studying group theory, and it distinguishes from school algebra so much!! All content on my language (Russian) is so complex and formal, that it is almost impossible to comprehend how algebra works. But this videos saved me! Thanks for all your work, please, continue to make videos!

[@crabsynth3480]

Just Found this Channel... I'm Hooked! Great Quality Content !

[@jinjutokkistudios6549]

I love you. You made algebra sound so easy to me and Ive been crying for two years to undestand it

[@semiconductorschematicdesign]

Finally I got to know about Groups, so good explanation,

[@mmanagengcaroline3219]

wow thank you ....a whole week of attending lectures covered simply in an 11 min video....amazing 🔥🔥🔥🔥

[@Sivenruot]

Great video ! I would have liked to have you as my teacher back in school ! Keep going on this amazing series of Math videos ! ❤️ from France !

[@studying8011]

Finally I found a teacher who believes more in understanding basics rather than rote learning Namaste

[@user-mb2dj9wf8z]

No one: What‘s algebra Me: when you draw your root out of an unknown in the morning

[@naveenchandrakumar480]

I have spent countless hours in the jungle of mathematics, to finally arrive at this. Really, commending job

[@غبيفيالرياضيات]

Abstract algebra♡♡ keep on post videos i love it

[@emilymylove5806]

Support on patreon..!!

[@krystoferhernandez7690]

You are incredible at explaining abstract. I learned more in this 11: 14-minute video than I did in two 50 minute lectures at my school. Thank you for making these types of videos. THANK YOU

[@manaskumarpandey6837]

please upload videos on MULTIVARIABLE CALCULUS and VECTOR CALCULUS. ..... THANKS.....

[@tomhu7917]

waiting for you to upload these material to enable us easy learning

[@hao-jen1887]

I never thought I could learn advanced mathematics in such an engaging way

[@caio868]

Amazing teaching abilities. It's incredible how one-hour spending reading a famous math textbook did not teach me the same amount of information as you did in this video. Would you know why textbooks are not done in this way? Since authors are so smart, why can't they realize books should be written in this way? It's really an honest question if you don't mind answering.

[@RadicalCaveman]

It's a mystery that the best mathematical minds are unable to penetrate...

[@JLKeener77]

I agree! It’s almost like mathematicians take pride in making their textbooks as dense and opaque as possible. Math is hard enough! I don’t know why math books can’t break the concepts down in both an informal way and then with more rigor *after* the concept has first been simply explained. These videos are absolutely outstanding!

[@cyrus_momo]

This is actually so useful for the a level further add pure segment, not much teacher can actually explain it properly

[@amydebuitleir]

Q: What's purple and commutes? A: An Abelian grape.

[@Jared7873]

🍇

[@imppie3754]

XD

[@Rhaenyra-k9k]

U r a legend ma'am , what an explanation , fantastic This is what we needed (quality education) Thank u ma'am

[@opchaudhary1188]

extra ordinary videos very helpful

[@Socratica]

Sign up to our email list to be notified when we release more Abstract Algebra content: snu.socratica.com/abstract-algebra

[@randyzeitman1354]

1) "We call this transformation R-squared as we applied it twice." Why R-squared and not 2*R. Is it because "x" is a dimensional operator and not scaler as "•" the dot operator? In other words, if the triangle was moved to the right or left, 'translated', would that be -2X (left) and 2X (right)? And if the triangle was moved to the up or down would that be translation onto the -i (imaginary vs. real number) axis? -2IX and 2IX? 2) If one can rotate it 180° about a vertical axis why not 'swing' 360° it about a horizontal axis positioned at the apex?

[@michaelsommers2356]

It's called R^2 because you are applying the R operation twice: RR, or R times R, or R squared.

[@Hutt-px1jx]

Michael Sommers Why does applying the rotation twice have to mean R*R? Why can’t we understand stand it as R+R (i.e. 2*R).

[@danielmartchenkov7437]

@@Hutt-px1jx its because we call the operation on that group "multiplication" so applying x and then y is written x times y so r times r is r^2. Its just convention. If we called it "plus" like we did for clock arithmetic it would be 2*r.

[@randyzeitman1354]

@@danielmartchenkov7437 If the triangle was moved to the right or left, 'linearly translated', would that be -2X (left) and 2X (right)? And if the triangle was moved to the up or down would that be translation onto the -i (imaginary vs. real number) axis? -2IX and 2IX? If it's just convention then call it cosine. Clock arithmetic would seem to be "plus" because it's a scaler and rotational translation has to be multi-dimensional so it's multi-plied ... no? ... 4x5=20 (sq. units) vs. 4•5=20 (scaler).

[@danielmartchenkov7437]

​@@randyzeitman1354 You can also see it's not normal addition nor normal multiplication since in the symmetries of a triangle example it's not even commutative. Also the complex extension of multiplication doesn't apply since calling it multiplication is just a convention. Like the examples with dot products, cross products you bring up-you wouldn't extend those to the complex plane, even though they're still called "multiplication". And about calling it cosine.. the reason it's given the specific name of addition or multiplication is because of how it was discovered: the operation on the modular group on the integers very closely resembles addition, and a common convention is to associate composition of "actions" (in this case the action of doing something to the triangle) to multiplication. I'm not entirely sure where that started, but you can see it in linear algebra when you consider matrix "multiplication" (even though it is technically the composition of the linear functions that each matrix represents, which are transformations (action) of the vector space. And again! not commutative). In fact, similarly to the matrices, the elements of a symmetric group (which in this case were the rotations and flips) are often denoted by functions. So the rotation clockwise would be the function that takes 1->2, 2->3, 3->1, if you imagine how the triangle with the numbered corners looks after that transformation. Then, doing an reflection, r, followed by a flip, f, is the function f(r), which takes the top-most point to f(r(1)), etc. This means that, like with the matrices in LA, the operation is in fact *function composition*, but when we intuitively consider it to be doing certain actions one after the other, we just write it as multiplication.

[@NickDorogavtsev]

Thanks

[@rheetuparnogogoi8780]

In the end, in the definition you gave the identity should be mentioned prior to the inverse. Without the existence of identity there is no reason of speaking about inverse

[@shubhoroy8438]

Actually a nifty exercise to appreciate how crucial associativity is in the business of groups is the fact that courtesy associativity the left and right identity element of group are the one and the same, and the left and right inverses of any element of the group are the one and the same. Good work, keep it up :)

[@PierreSoubourou]

Doesn't mod stand for modulo (or sometimes modulus)?

[@steliostoulis1875]

Pierre Soubourou yes

[@Calm_Energy]

modulus is (mathematics) the base with respect to which a congruence is computed while modulo is (computing) the operation or function that returns the remainder of one number divided by another. I also discovered there is a difference between mod and remainder when you divide negative numbers! 🙂

[@suspendedsuplexchannel1000]

Pierre Soubourou it is called "module" actuallu

[@RolandHutchinson]

Yes, it's "modulo" -- the Latin ablative case form of "modulus". The meaning of the ablative case form is something like "using the modulus". So in 7 ≡ 2 (mod 5), pronounced "seven is congruent to two mod five", "mod 5" means "using the modulus 5", or "five being the modulus". This is definitely a (minor) error in the script of the video.

[@hainguyen9148]

Although I already watched many other videos, I still have not understood the Group Definition until I watch this video, Thanks a lot, you are a good teacher.

[@zahretallotos3724]

Please we want a lessons about "Nilpotent Group or ideal" and and about "sylow " 😭💔🙏🙏

[@harryh212]

Well done for this video. My lecturer and textbook fail to lay this out so well. They just scramble a bunch of these points together with no explanation making something so simple feel impossible. So thanks for the clarity.

[@pedronpp]

You are perfect for the next COSMOS! Carl Sagan, Niel deGrasse Tyson and you!

[@bckzilla]

Yep - and her Python robot counterpart.

[@James-qb6cs]

I've tried watching other group theory lectures and yours are by far the most engaging and understandable!

[@somenn.s3977]

Add some videos on Real analysis and calculus.

[@Socratica]

More math videos coming soon!

[@mrugeshdhroovji]

Thank you so much for explaining this so good never had a teacher who explained it so good

[@JyotiYadav-md4wy]

why we have no vertical rotation in triangles?

[@b43xoit]

Because that can be made of the operations she did include?

[@rmirabal12000]

Congratulations,you are doing a great service to humans interested in math.

[@Zebedex]

Cortana became human?

[@claudefazio]

You're an excellent teacher! The clarity of your explanations is astounding.

[@jessyjonathas7296]

I love your job! I will do everything to follow you! But I have my limits and I miss the Portuguese Socratica. Please look for sponsorship (example: DUOLINGO) and re-invest in the language content of our good old Brazilian literary (João Guimarães Rosa, Machado de Assis, Cecília Meireles). I and other MILLIONS of Brazilians, sometimes limited by our labor and financial conditions, want many stools without power to follow the EDUCATION you sponsored. Help us to help you come to our sad reality of not often recerber a BILINGUAL education. I and other MILLIONS of internauts are grateful, may God be with you in the struggle for a quality education in this NATION! (Brazil)

[@robharwood3538]

Wow! Best complete and comprehensible definition of groups I've ever come across. Thanks for this video, but especially for your approach at being complete but simultaneously comprehensible and clear. Considering all the arcane confusion out there, yours seems to be a rare and important skill.

[@kommareddyrahulreddy2687]

just thinking how much effort you have put into the video.Hats off guys...

[@tareksaad1393]

Math is NOT boring .... says a 35 years old who failed math up until 12th grade ...but things changed when 2 things happened! Watching your videos , and bitcoin... perfect complements to ignite the love of math!

[@Superstino]

You gotta be kidding me. I have an exam about this in two days! Your timing couldn't have been any more perfect. Thanks!

[@eldeee143]

I'm watching this now prior to my comprehensive examination in my Master's degree. Cheers and thank you for your hard work in producing videos that are understandable to an extent. :)

[@stefanosmandalas6599]

Holly molly! I expected this playlist to be really hard but it turned out having the perfect level of difficulty! I love what Socratica is doing, really focusing on clarity above all, regardless the subject. Keep up the good work :)

[@HaleemaSadiya-ib1hq]

This video was really very helpful for any undergraduate as myself. Thank u so much for providing us with such a well defined and well organised explaination.

[@piyushnaithani7689]

oh my god, finally I find something great on youtube for a long time, Greate to see you artist..

[@jeremylaughery2555]

This is a wonderful resource for learning abstract algebra. I will rely heavily on abstract or modern algebra for my work on prime factorization methods. I have developed a fact that all consecutive primes have a finite period associated with an additive inverse and have even symmetry. I was able to use discrete Fourier transforms to describe the n-th explicit odd integer for the first several primes. These counting functions are far more complicated than the fundamental 2n+1 counting structure. Abstract algebra helped me think in more general terms to develop very complex counting algorithms for prime factorization. As the primes grow, so does the period associated with them. Therefore, we need to compress the elements of the group so that we can work with the group. Abstract algebra to the rescue! We need to be able to move around freely in a ratio-space while testing potential prime factors. Homomorphisms to the rescue! Abstract algebra will help solve the RSA problem! Finally, Lenstra's ECM can be shown to belong to a quadratic field through equivalence classes, another brilliant idea from abstract algebra!

[@oliviac5650]

I'm 10 yr and I love math but I can't find anything hard enough for me. Thank you for helping me learn abstract algebra with examples I understand, and explaining things clearly and thoroughly

[@gswovoka6286]

Thanks for this. I vaguely remember taking Abstract Algebra back in the 60s. I could not however remember any detail. It all came flooding back when you defined a Group.

[@noahshirey121]

I have been blown away by every single one of your videos I have watched. This channel is definitely a hidden gem. Thank you for your content!